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Well-order
Examples and counterexamples
Reals
The standard ordering ≦ of any real interval is not a well ordering, since, for example, the open interval (0, 1) ⊆ [0,1] does not contain a least element.
From the ZFC axioms of set theory (including the axiom of choice) one can show that there is a well order of the reals.
Nonetheless, it is possible to show that the ZFC+GCH axioms alone are not sufficient to prove the existence of a definable (by a formula) well order of the reals.[1]
However it is consistent with ZFC that a definable well ordering of the reals exists-for example, it is consistent with ZFC that V=L, and it follows from ZFC+V=L that a particular formula well orders the reals, or indeed any set.
References
1＾S. Feferman Some applications of the notions of forcing and generic sets Fundamenta Mathematicae (1964)
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S. Feferman Some applications of the notions of forcing and generic sets Fundamenta Mathematicae (1964)
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P1
The most interesting of these are the following:
(1) No set-theoreticallydefinable well-ordering of the continuum can be proved to exist fromthe Zermelo-Fraenkel axioms together with the axiom of choice andthe generalized continuum hypothesis.
P9
4.11 THEOREM. If s=1 there is no set-theoretically definable well-ordering of the continuum in M*.
Proof. What comes to the same thing, there is no formula F(X, Y)of L which establishes a well-ordering relation in the set of all subsetsFundamenta Mathematicae, T. LVI 略
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